Bounds on multiplicities of symmetric pairs of finite groups

Let $\mathrm{\Gamma }$ be a finite group, let $\theta$ be an involution of $\mathrm{\Gamma }$, and let $\rho$ be an irreducible complex representation of $\mathrm{\Gamma }$. We bound $\dim {\rho }^{{\mathrm{\Gamma }}^{\theta }}$ in terms of the smallest dimension of a faithful ${\mathbb{F}}_p$-representation of $\mathrm{\Gamma }\mathrm{/}Ra{d}_p(\mathrm{\Gamma })$, where $p$ is any odd prime and $Ra{d}_p(\mathrm{\Gamma })$ is the maximal normal $p$-subgroup of $\mathrm{\Gamma }$. This implies, in particular, that if $\mathbf{G}$ is a group scheme over $\mathbb{Z}$ and $\theta$ is an involution of $\mathbf{G}$, then the multiplicity of any irreducible representation in $C^\infty \left( $\mathbf{G}$($\mathbb{Z}$_p)/ $\mathbf{G}$ ^{\theta}($\mathbb{Z}$_p) \right)$ is bounded, uniformly in $p$.